In mathematics, the max–min inequality is as follows:
For any function
When equality holds one says that , , and satisfies a strong max–min property (or a saddle-point property). The example function illustrates that the equality does not hold for every function.
A theorem giving conditions on , , and which guarantee the saddle point property is called a minimax theorem.
Proof Define For all , we get for all by definition of the infimum being a lower bound. Next, for all , for all by definition of the supremum being an upper bound. Thus, for all and , making an upper bound on for any choice of . Because the supremum is the least upper bound, holds for all . From this inequality, we also see that is a lower bound on . By the greatest lower bound property of infimum, . Putting all the pieces together, we get
which proves the desired inequality.
References
See also Minimax theorem